Abstract
Uncertainty is an intrinsic property of rock engineering because of the complicated geology conditions, rock failure mechanism ambiguity, and the nonlinear mechanical behavior of surrounding rock mass. We developed a novel framework to handle the uncertainty by combing the Sparse polynomial chaotic expansion (SPCE), numerical model, and reliability method. The SPCE model was used to map the complex relationship between the response of the surrounding rock mass and its uncertainty. The first-order reliability method (FORM) evaluated the reliability index and failure probability. Based on the SPCE model and FORM, a simple global optimization algorithm (SHGO) seeks design points and corresponding reliability indexes. A circular tunnel verified the developed framework with a close-form solution. The reliability index, design point, and failure probability were in excellent agreement with the FORM and Monte Carlo simulation. This indicated that the SPCE model could be used as a surrogate model for the analytical solution to approximate the tunnel response (including deformation and size of the plastic zone). Then, the developed framework was employed in a horseshoe tunnel by combing with the numerical model. The results further proved that the developed framework is feasible and effective for handling uncertainty in rock engineering. Furthermore, the developed framework is effective, efficient, and accurate for reliability analysis and provides a helpful tool to approximate the response of rock structure to avoid the time-consuming numerical model in practical rock engineering.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Au SK, Beck JL (1999) A new adaptive importance sampling scheme for reliability calculations. Structural Safety 21(2):135–138, DOI: https://doi.org/10.1016/S0167-4730(99)00014-4
Blatman G, Sudret B (2010) An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Probabilistic Engineering Mechanics 25(2):183–197, DOI: https://doi.org/10.1016/j.probengmech.2009.10.003
Blatman G, Sudret B (2011) Adaptive sparse polynomial chaos expansion based on least angle regression. Journal of Computational Physics 230(6):2345–2367, DOI: https://doi.org/10.1016/j.jcp.2010.12.021
Cho SE (2009) Probabilistic stability analyses of slopes using the ANN-based response surface. Computers and Geotechnics 36(5):787–797, DOI: https://doi.org/10.1016/j.compgeo.2009.01.003
Chowdhury R, Rao BN (2010) Probabilistic stability assessment of slopes using high dimensional model representation. Computers and Geotechnics 37(7):876–884, DOI: https://doi.org/10.1016/j.compgeo.2010.07.007
Deng J, Gu D, Li X, Yue ZQ (2005) Structural reliability analysis for implicit performance functions using artificial neural network. Structural Safety 27(1):25–48, DOI: https://doi.org/10.1016/j.strusafe.2004.03.004
Ditlevsen O, Madsen HO (1996) Structural reliability methods. John Wiley and Sons, Inc., Chichester, England, 163–177
Duncan Fama ME (1993) Numerical modeling of yield zones in weak rocks. In: Hudson JA, editor. Comprehensive Rock Engineering 2:49–75, DOI: https://doi.org/10.1016/B978-0-08-040615-2.50009-5
Endres SC, Sandrock C, Focke WW (2018) A simplicial homology algorithm for lipschitz optimisation. Journal of Global Optimization 72:181–217, DOI: https://doi.org/10.1007/s10898-018-0645-y
Fellin W, King J, Kirsch A, Oberguggenberger M (2010) Uncertainty modelling and sensitivity analysis of tunnel face stability. Structural Safety 32(6):402–410, DOI: https://doi.org/10.1016/j.strusafe.2010.06.001
Griffiths DV, Huang J, Gordon AF (2011) Probabilistic infinite slope analysis. Computers and Geotechnics 38(4):577–584, DOI: https://doi.org/10.1016/j.compgeo.2011.03.006
Hoek E (1998) Reliability of Hoek-Brown estimates of rock mass properties and their impact on design. International Journal of Rock Mechanics and Mining Sciences 35(1):63–68, DOI: https://doi.org/10.1016/S0148-9062(97)00314-8
Hsu WC, Ching JY (2010) Evaluating small failure probabilities of multiple limit states by parallel subset simulation. Probabilistic Engineering Mechanics 25(3):291–304, DOI: https://doi.org/10.1016/j.probengmech.2010.01.003
Jing L, Hudson JA (2002) Numerical methods in rock mechanics. International Journal of Rock Mechanics and Mining Sciences 39(4):409–427, DOI: https://doi.org/10.1016/S1365-1609(02)00065-5
Kang F, Li JJ (2016) Artificial bee colony algorithm optimized support vector regression for system reliability analysis of slopes. Journal of Computing in Civil Engineering 30(3):04015040, DOI: https://doi.org/10.1061/(ASCE)CP.1943-5487.0000514
Li DQ, Zheng D, Cao ZJ, Tang XS, Phoon KK (2016) Response surface methods for slope reliability analysis: Review and comparison. Engineering Geology 203:3–14, DOI: https://doi.org/10.1016/j.enggeo.2015.09.003
Low BK (2004) Reliability analysis using object-oriented constrained optimization. Structural Safety 26(1):69–89, DOI: https://doi.org/10.1016/S0167-4730(03)00023-7
Low BK, Tang WH (2007) Efficient spreadsheet algorithm for firstorder reliability method. Journal of Engineering Mechanics 133(12): 1378–1387, DOI: https://doi.org/10.1061/(ASCE)0733-9399(2007)133:12(1378)
Luo XF, Li X, Zhou J, Cheng T (2012) A Kriging-based hybrid optimization algorithm for slope reliability analysis. Structural Safety 34(1):401–406, DOI: https://doi.org/10.1016/j.strusafe.2011.09.004
Lv Q, Low BK (2011) Probabilistic analysis of underground rock excavations using response surface method and SORM. Computers and Geotechnics 38(8):1008–1021, DOI: https://doi.org/10.1016/j.compgeo.2011.07.003
Panthi KK, Nilsen B (2007) Uncertainty analysis of tunnel squeezing for two tunnel cases from Nepal Himalaya. International Journal of Rock Mechanics and Mining Sciences 44(1):67–76, DOI: https://doi.org/10.1016/j.ijrmms.2006.04.013
Sperner E (1928) Neuer beweis für die invarianz der dimensionszahl und des gebietes. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 6(1):265–272, DOI: https://doi.org/10.1007/BF02940617
Tan XH, Bi WH, Hou XL, Wang W (2011) Reliability analysis using radial basis function networks and support vector machines. Computers and Geotechnics 38(2):178–186, DOI: https://doi.org/10.1016/j.compgeo.2010.11.002
Tiwari G, Pandit B, Madhavi G, Latha GL, Babu S (2017) Probabilistic analysis of tunnels considering uncertainty in peak and post-peak strength parameters. Tunnelling and Underground Space Technology 70:375–387, DOI: https://doi.org/10.1016/j.tust.2017.09.013
Xiu D, Karniadakis GE (2002) The wiener-askey polynomial chaos for stochastic differential equations. Siam J. Sci. Comput. 24(2):619–644, DOI: https://doi.org/10.1137/S1064827501387826
Zhang L, Wang M, Zhao H, Chang X (2022) Uncertainty quantification for the mechanical behavior of fully grouted rockbolts subjected to pull-out tests. Computers and Geotechnics 145:104665, DOI: https://doi.org/10.1016/j.compgeo.2022.104665
Zhao H (2008) Slope reliability analysis using a support vector machine. Computers and Geotechnics 35(3):459–467, DOI: https://doi.org/10.1016/j.compgeo.2007.08.002
Zhao H, Chen B, Li S (2021a) Determination of geomaterial mechanical parameters based on back analysis and reduced-order model. Computers and Geotechnics 132:104013, DOI: https://doi.org/10.1016/j.compgeo.2021.104013
Zhao H, Chen B, Li S, Li Z, Zhu C (2021b) Updating the models and uncertainty of mechanical parameters for rock tunnels using Bayesian inference. Geoscience Frontiers 12(5):101198, DOI: https://doi.org/10.1016/j.gsf.2021.101198
Zhao H, Li S, Ru Z (2017) Adaptive reliability analysis approach based on SVM and its application. Applied Mathematical Modelling 44:508–522, DOI: https://doi.org/10.1016/j.apm.2017.02.020
Zhao H, Li S, Zhu C (2021c) Uncertainty analysis of rock tunnel based on fractional moment and dimensional reduction method. International Journal of Rock Mechanics and Mining Sciences 139:104656, DOI: https://doi.org/10.1016/j.ijrmms.2021.104656
Zhao YG, Ono T (1999) A general procedure for first/second-order reliability method (FORM/SORM). Structural Safety 21(2):95–112, DOI: https://doi.org/10.1016/S0167-4730(99)00008-9
Zhao H, Ru Z, Chang X, Yin S, Li S (2014) Reliability analysis of tunnel using a least square support vector machine. Tunnelling and Underground Space Technology 41:14–23, DOI: https://doi.org/10.1016/j.tust.2013.11.004
Acknowledgments
The authors gratefully acknowledge the financial support provided by the National Key Laboratory of Geomechanics and Geotechnical Engineering (Grant No. Z020006).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhao, H., Wang, M., Chen, B. et al. Sparse Polynomial Chaotic Expansion for Uncertainty Analysis of Tunnel Stability. KSCE J Civ Eng 26, 3992–4003 (2022). https://doi.org/10.1007/s12205-022-2099-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12205-022-2099-5